1200 lines
32 KiB
Python
1200 lines
32 KiB
Python
"""Imported from the recipes section of the itertools documentation.
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All functions taken from the recipes section of the itertools library docs
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[1]_.
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Some backward-compatible usability improvements have been made.
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.. [1] http://docs.python.org/library/itertools.html#recipes
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"""
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import math
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import operator
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from collections import deque
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from collections.abc import Sized
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from functools import lru_cache, partial
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from itertools import (
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chain,
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combinations,
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compress,
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count,
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cycle,
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groupby,
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islice,
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product,
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repeat,
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starmap,
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tee,
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zip_longest,
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)
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from random import randrange, sample, choice
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from sys import hexversion
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__all__ = [
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'all_equal',
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'batched',
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'before_and_after',
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'consume',
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'convolve',
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'dotproduct',
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'first_true',
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'factor',
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'flatten',
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'grouper',
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'is_prime',
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'iter_except',
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'iter_index',
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'loops',
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'matmul',
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'ncycles',
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'nth',
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'nth_combination',
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'padnone',
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'pad_none',
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'pairwise',
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'partition',
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'polynomial_eval',
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'polynomial_from_roots',
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'polynomial_derivative',
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'powerset',
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'prepend',
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'quantify',
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'reshape',
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'random_combination_with_replacement',
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'random_combination',
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'random_permutation',
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'random_product',
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'repeatfunc',
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'roundrobin',
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'sieve',
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'sliding_window',
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'subslices',
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'sum_of_squares',
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'tabulate',
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'tail',
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'take',
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'totient',
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'transpose',
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'triplewise',
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'unique',
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'unique_everseen',
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'unique_justseen',
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]
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_marker = object()
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# zip with strict is available for Python 3.10+
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try:
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zip(strict=True)
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except TypeError:
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_zip_strict = zip
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else:
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_zip_strict = partial(zip, strict=True)
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# math.sumprod is available for Python 3.12+
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_sumprod = getattr(math, 'sumprod', lambda x, y: dotproduct(x, y))
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def take(n, iterable):
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"""Return first *n* items of the iterable as a list.
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>>> take(3, range(10))
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[0, 1, 2]
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If there are fewer than *n* items in the iterable, all of them are
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returned.
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>>> take(10, range(3))
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[0, 1, 2]
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"""
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return list(islice(iterable, n))
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def tabulate(function, start=0):
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"""Return an iterator over the results of ``func(start)``,
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``func(start + 1)``, ``func(start + 2)``...
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*func* should be a function that accepts one integer argument.
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If *start* is not specified it defaults to 0. It will be incremented each
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time the iterator is advanced.
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>>> square = lambda x: x ** 2
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>>> iterator = tabulate(square, -3)
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>>> take(4, iterator)
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[9, 4, 1, 0]
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"""
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return map(function, count(start))
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def tail(n, iterable):
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"""Return an iterator over the last *n* items of *iterable*.
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>>> t = tail(3, 'ABCDEFG')
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>>> list(t)
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['E', 'F', 'G']
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"""
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# If the given iterable has a length, then we can use islice to get its
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# final elements. Note that if the iterable is not actually Iterable,
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# either islice or deque will throw a TypeError. This is why we don't
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# check if it is Iterable.
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if isinstance(iterable, Sized):
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yield from islice(iterable, max(0, len(iterable) - n), None)
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else:
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yield from iter(deque(iterable, maxlen=n))
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def consume(iterator, n=None):
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"""Advance *iterable* by *n* steps. If *n* is ``None``, consume it
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entirely.
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Efficiently exhausts an iterator without returning values. Defaults to
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consuming the whole iterator, but an optional second argument may be
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provided to limit consumption.
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>>> i = (x for x in range(10))
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>>> next(i)
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0
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>>> consume(i, 3)
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>>> next(i)
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4
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>>> consume(i)
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>>> next(i)
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Traceback (most recent call last):
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File "<stdin>", line 1, in <module>
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StopIteration
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If the iterator has fewer items remaining than the provided limit, the
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whole iterator will be consumed.
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>>> i = (x for x in range(3))
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>>> consume(i, 5)
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>>> next(i)
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Traceback (most recent call last):
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File "<stdin>", line 1, in <module>
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StopIteration
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"""
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# Use functions that consume iterators at C speed.
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if n is None:
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# feed the entire iterator into a zero-length deque
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deque(iterator, maxlen=0)
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else:
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# advance to the empty slice starting at position n
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next(islice(iterator, n, n), None)
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def nth(iterable, n, default=None):
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"""Returns the nth item or a default value.
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>>> l = range(10)
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>>> nth(l, 3)
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3
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>>> nth(l, 20, "zebra")
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'zebra'
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"""
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return next(islice(iterable, n, None), default)
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def all_equal(iterable, key=None):
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"""
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Returns ``True`` if all the elements are equal to each other.
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>>> all_equal('aaaa')
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True
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>>> all_equal('aaab')
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False
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A function that accepts a single argument and returns a transformed version
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of each input item can be specified with *key*:
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>>> all_equal('AaaA', key=str.casefold)
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True
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>>> all_equal([1, 2, 3], key=lambda x: x < 10)
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True
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"""
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iterator = groupby(iterable, key)
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for first in iterator:
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for second in iterator:
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return False
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return True
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return True
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def quantify(iterable, pred=bool):
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"""Return the how many times the predicate is true.
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>>> quantify([True, False, True])
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2
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"""
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return sum(map(pred, iterable))
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def pad_none(iterable):
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"""Returns the sequence of elements and then returns ``None`` indefinitely.
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>>> take(5, pad_none(range(3)))
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[0, 1, 2, None, None]
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Useful for emulating the behavior of the built-in :func:`map` function.
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See also :func:`padded`.
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"""
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return chain(iterable, repeat(None))
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padnone = pad_none
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def ncycles(iterable, n):
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"""Returns the sequence elements *n* times
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>>> list(ncycles(["a", "b"], 3))
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['a', 'b', 'a', 'b', 'a', 'b']
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"""
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return chain.from_iterable(repeat(tuple(iterable), n))
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def dotproduct(vec1, vec2):
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"""Returns the dot product of the two iterables.
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>>> dotproduct([10, 10], [20, 20])
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400
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"""
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return sum(map(operator.mul, vec1, vec2))
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def flatten(listOfLists):
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"""Return an iterator flattening one level of nesting in a list of lists.
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>>> list(flatten([[0, 1], [2, 3]]))
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[0, 1, 2, 3]
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See also :func:`collapse`, which can flatten multiple levels of nesting.
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"""
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return chain.from_iterable(listOfLists)
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def repeatfunc(func, times=None, *args):
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"""Call *func* with *args* repeatedly, returning an iterable over the
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results.
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If *times* is specified, the iterable will terminate after that many
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repetitions:
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>>> from operator import add
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>>> times = 4
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>>> args = 3, 5
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>>> list(repeatfunc(add, times, *args))
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[8, 8, 8, 8]
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If *times* is ``None`` the iterable will not terminate:
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>>> from random import randrange
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>>> times = None
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>>> args = 1, 11
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>>> take(6, repeatfunc(randrange, times, *args)) # doctest:+SKIP
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[2, 4, 8, 1, 8, 4]
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"""
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if times is None:
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return starmap(func, repeat(args))
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return starmap(func, repeat(args, times))
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def _pairwise(iterable):
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"""Returns an iterator of paired items, overlapping, from the original
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>>> take(4, pairwise(count()))
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[(0, 1), (1, 2), (2, 3), (3, 4)]
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On Python 3.10 and above, this is an alias for :func:`itertools.pairwise`.
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"""
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a, b = tee(iterable)
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next(b, None)
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return zip(a, b)
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try:
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from itertools import pairwise as itertools_pairwise
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except ImportError:
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pairwise = _pairwise
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else:
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def pairwise(iterable):
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return itertools_pairwise(iterable)
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pairwise.__doc__ = _pairwise.__doc__
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class UnequalIterablesError(ValueError):
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def __init__(self, details=None):
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msg = 'Iterables have different lengths'
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if details is not None:
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msg += (': index 0 has length {}; index {} has length {}').format(
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*details
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)
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super().__init__(msg)
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def _zip_equal_generator(iterables):
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for combo in zip_longest(*iterables, fillvalue=_marker):
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for val in combo:
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if val is _marker:
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raise UnequalIterablesError()
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yield combo
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def _zip_equal(*iterables):
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# Check whether the iterables are all the same size.
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try:
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first_size = len(iterables[0])
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for i, it in enumerate(iterables[1:], 1):
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size = len(it)
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if size != first_size:
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raise UnequalIterablesError(details=(first_size, i, size))
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# All sizes are equal, we can use the built-in zip.
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return zip(*iterables)
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# If any one of the iterables didn't have a length, start reading
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# them until one runs out.
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except TypeError:
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return _zip_equal_generator(iterables)
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def grouper(iterable, n, incomplete='fill', fillvalue=None):
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"""Group elements from *iterable* into fixed-length groups of length *n*.
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>>> list(grouper('ABCDEF', 3))
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[('A', 'B', 'C'), ('D', 'E', 'F')]
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The keyword arguments *incomplete* and *fillvalue* control what happens for
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iterables whose length is not a multiple of *n*.
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When *incomplete* is `'fill'`, the last group will contain instances of
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*fillvalue*.
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>>> list(grouper('ABCDEFG', 3, incomplete='fill', fillvalue='x'))
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[('A', 'B', 'C'), ('D', 'E', 'F'), ('G', 'x', 'x')]
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When *incomplete* is `'ignore'`, the last group will not be emitted.
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>>> list(grouper('ABCDEFG', 3, incomplete='ignore', fillvalue='x'))
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[('A', 'B', 'C'), ('D', 'E', 'F')]
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When *incomplete* is `'strict'`, a subclass of `ValueError` will be raised.
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>>> it = grouper('ABCDEFG', 3, incomplete='strict')
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>>> list(it) # doctest: +IGNORE_EXCEPTION_DETAIL
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Traceback (most recent call last):
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...
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UnequalIterablesError
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"""
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args = [iter(iterable)] * n
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if incomplete == 'fill':
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return zip_longest(*args, fillvalue=fillvalue)
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if incomplete == 'strict':
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return _zip_equal(*args)
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if incomplete == 'ignore':
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return zip(*args)
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else:
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raise ValueError('Expected fill, strict, or ignore')
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def roundrobin(*iterables):
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"""Yields an item from each iterable, alternating between them.
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>>> list(roundrobin('ABC', 'D', 'EF'))
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['A', 'D', 'E', 'B', 'F', 'C']
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This function produces the same output as :func:`interleave_longest`, but
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may perform better for some inputs (in particular when the number of
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iterables is small).
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"""
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# Algorithm credited to George Sakkis
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iterators = map(iter, iterables)
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for num_active in range(len(iterables), 0, -1):
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iterators = cycle(islice(iterators, num_active))
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yield from map(next, iterators)
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def partition(pred, iterable):
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"""
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Returns a 2-tuple of iterables derived from the input iterable.
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The first yields the items that have ``pred(item) == False``.
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The second yields the items that have ``pred(item) == True``.
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>>> is_odd = lambda x: x % 2 != 0
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>>> iterable = range(10)
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>>> even_items, odd_items = partition(is_odd, iterable)
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>>> list(even_items), list(odd_items)
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([0, 2, 4, 6, 8], [1, 3, 5, 7, 9])
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If *pred* is None, :func:`bool` is used.
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>>> iterable = [0, 1, False, True, '', ' ']
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>>> false_items, true_items = partition(None, iterable)
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>>> list(false_items), list(true_items)
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([0, False, ''], [1, True, ' '])
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"""
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if pred is None:
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pred = bool
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t1, t2, p = tee(iterable, 3)
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p1, p2 = tee(map(pred, p))
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return (compress(t1, map(operator.not_, p1)), compress(t2, p2))
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def powerset(iterable):
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"""Yields all possible subsets of the iterable.
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>>> list(powerset([1, 2, 3]))
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[(), (1,), (2,), (3,), (1, 2), (1, 3), (2, 3), (1, 2, 3)]
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:func:`powerset` will operate on iterables that aren't :class:`set`
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instances, so repeated elements in the input will produce repeated elements
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in the output.
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>>> seq = [1, 1, 0]
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>>> list(powerset(seq))
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[(), (1,), (1,), (0,), (1, 1), (1, 0), (1, 0), (1, 1, 0)]
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For a variant that efficiently yields actual :class:`set` instances, see
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:func:`powerset_of_sets`.
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"""
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s = list(iterable)
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return chain.from_iterable(combinations(s, r) for r in range(len(s) + 1))
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def unique_everseen(iterable, key=None):
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"""
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Yield unique elements, preserving order.
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>>> list(unique_everseen('AAAABBBCCDAABBB'))
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['A', 'B', 'C', 'D']
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>>> list(unique_everseen('ABBCcAD', str.lower))
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['A', 'B', 'C', 'D']
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Sequences with a mix of hashable and unhashable items can be used.
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The function will be slower (i.e., `O(n^2)`) for unhashable items.
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Remember that ``list`` objects are unhashable - you can use the *key*
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parameter to transform the list to a tuple (which is hashable) to
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avoid a slowdown.
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>>> iterable = ([1, 2], [2, 3], [1, 2])
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>>> list(unique_everseen(iterable)) # Slow
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[[1, 2], [2, 3]]
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>>> list(unique_everseen(iterable, key=tuple)) # Faster
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[[1, 2], [2, 3]]
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Similarly, you may want to convert unhashable ``set`` objects with
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``key=frozenset``. For ``dict`` objects,
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``key=lambda x: frozenset(x.items())`` can be used.
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"""
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seenset = set()
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seenset_add = seenset.add
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seenlist = []
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seenlist_add = seenlist.append
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use_key = key is not None
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for element in iterable:
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k = key(element) if use_key else element
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try:
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if k not in seenset:
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seenset_add(k)
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yield element
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except TypeError:
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if k not in seenlist:
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seenlist_add(k)
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yield element
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def unique_justseen(iterable, key=None):
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"""Yields elements in order, ignoring serial duplicates
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>>> list(unique_justseen('AAAABBBCCDAABBB'))
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['A', 'B', 'C', 'D', 'A', 'B']
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>>> list(unique_justseen('ABBCcAD', str.lower))
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['A', 'B', 'C', 'A', 'D']
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"""
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if key is None:
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return map(operator.itemgetter(0), groupby(iterable))
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return map(next, map(operator.itemgetter(1), groupby(iterable, key)))
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def unique(iterable, key=None, reverse=False):
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"""Yields unique elements in sorted order.
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>>> list(unique([[1, 2], [3, 4], [1, 2]]))
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[[1, 2], [3, 4]]
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*key* and *reverse* are passed to :func:`sorted`.
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>>> list(unique('ABBcCAD', str.casefold))
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['A', 'B', 'c', 'D']
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>>> list(unique('ABBcCAD', str.casefold, reverse=True))
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['D', 'c', 'B', 'A']
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The elements in *iterable* need not be hashable, but they must be
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comparable for sorting to work.
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"""
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return unique_justseen(sorted(iterable, key=key, reverse=reverse), key=key)
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|
||
def iter_except(func, exception, first=None):
|
||
"""Yields results from a function repeatedly until an exception is raised.
|
||
|
||
Converts a call-until-exception interface to an iterator interface.
|
||
Like ``iter(func, sentinel)``, but uses an exception instead of a sentinel
|
||
to end the loop.
|
||
|
||
>>> l = [0, 1, 2]
|
||
>>> list(iter_except(l.pop, IndexError))
|
||
[2, 1, 0]
|
||
|
||
Multiple exceptions can be specified as a stopping condition:
|
||
|
||
>>> l = [1, 2, 3, '...', 4, 5, 6]
|
||
>>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError)))
|
||
[7, 6, 5]
|
||
>>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError)))
|
||
[4, 3, 2]
|
||
>>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError)))
|
||
[]
|
||
|
||
"""
|
||
try:
|
||
if first is not None:
|
||
yield first()
|
||
while 1:
|
||
yield func()
|
||
except exception:
|
||
pass
|
||
|
||
|
||
def first_true(iterable, default=None, pred=None):
|
||
"""
|
||
Returns the first true value in the iterable.
|
||
|
||
If no true value is found, returns *default*
|
||
|
||
If *pred* is not None, returns the first item for which
|
||
``pred(item) == True`` .
|
||
|
||
>>> first_true(range(10))
|
||
1
|
||
>>> first_true(range(10), pred=lambda x: x > 5)
|
||
6
|
||
>>> first_true(range(10), default='missing', pred=lambda x: x > 9)
|
||
'missing'
|
||
|
||
"""
|
||
return next(filter(pred, iterable), default)
|
||
|
||
|
||
def random_product(*args, repeat=1):
|
||
"""Draw an item at random from each of the input iterables.
|
||
|
||
>>> random_product('abc', range(4), 'XYZ') # doctest:+SKIP
|
||
('c', 3, 'Z')
|
||
|
||
If *repeat* is provided as a keyword argument, that many items will be
|
||
drawn from each iterable.
|
||
|
||
>>> random_product('abcd', range(4), repeat=2) # doctest:+SKIP
|
||
('a', 2, 'd', 3)
|
||
|
||
This equivalent to taking a random selection from
|
||
``itertools.product(*args, **kwarg)``.
|
||
|
||
"""
|
||
pools = [tuple(pool) for pool in args] * repeat
|
||
return tuple(choice(pool) for pool in pools)
|
||
|
||
|
||
def random_permutation(iterable, r=None):
|
||
"""Return a random *r* length permutation of the elements in *iterable*.
|
||
|
||
If *r* is not specified or is ``None``, then *r* defaults to the length of
|
||
*iterable*.
|
||
|
||
>>> random_permutation(range(5)) # doctest:+SKIP
|
||
(3, 4, 0, 1, 2)
|
||
|
||
This equivalent to taking a random selection from
|
||
``itertools.permutations(iterable, r)``.
|
||
|
||
"""
|
||
pool = tuple(iterable)
|
||
r = len(pool) if r is None else r
|
||
return tuple(sample(pool, r))
|
||
|
||
|
||
def random_combination(iterable, r):
|
||
"""Return a random *r* length subsequence of the elements in *iterable*.
|
||
|
||
>>> random_combination(range(5), 3) # doctest:+SKIP
|
||
(2, 3, 4)
|
||
|
||
This equivalent to taking a random selection from
|
||
``itertools.combinations(iterable, r)``.
|
||
|
||
"""
|
||
pool = tuple(iterable)
|
||
n = len(pool)
|
||
indices = sorted(sample(range(n), r))
|
||
return tuple(pool[i] for i in indices)
|
||
|
||
|
||
def random_combination_with_replacement(iterable, r):
|
||
"""Return a random *r* length subsequence of elements in *iterable*,
|
||
allowing individual elements to be repeated.
|
||
|
||
>>> random_combination_with_replacement(range(3), 5) # doctest:+SKIP
|
||
(0, 0, 1, 2, 2)
|
||
|
||
This equivalent to taking a random selection from
|
||
``itertools.combinations_with_replacement(iterable, r)``.
|
||
|
||
"""
|
||
pool = tuple(iterable)
|
||
n = len(pool)
|
||
indices = sorted(randrange(n) for i in range(r))
|
||
return tuple(pool[i] for i in indices)
|
||
|
||
|
||
def nth_combination(iterable, r, index):
|
||
"""Equivalent to ``list(combinations(iterable, r))[index]``.
|
||
|
||
The subsequences of *iterable* that are of length *r* can be ordered
|
||
lexicographically. :func:`nth_combination` computes the subsequence at
|
||
sort position *index* directly, without computing the previous
|
||
subsequences.
|
||
|
||
>>> nth_combination(range(5), 3, 5)
|
||
(0, 3, 4)
|
||
|
||
``ValueError`` will be raised If *r* is negative or greater than the length
|
||
of *iterable*.
|
||
``IndexError`` will be raised if the given *index* is invalid.
|
||
"""
|
||
pool = tuple(iterable)
|
||
n = len(pool)
|
||
if (r < 0) or (r > n):
|
||
raise ValueError
|
||
|
||
c = 1
|
||
k = min(r, n - r)
|
||
for i in range(1, k + 1):
|
||
c = c * (n - k + i) // i
|
||
|
||
if index < 0:
|
||
index += c
|
||
|
||
if (index < 0) or (index >= c):
|
||
raise IndexError
|
||
|
||
result = []
|
||
while r:
|
||
c, n, r = c * r // n, n - 1, r - 1
|
||
while index >= c:
|
||
index -= c
|
||
c, n = c * (n - r) // n, n - 1
|
||
result.append(pool[-1 - n])
|
||
|
||
return tuple(result)
|
||
|
||
|
||
def prepend(value, iterator):
|
||
"""Yield *value*, followed by the elements in *iterator*.
|
||
|
||
>>> value = '0'
|
||
>>> iterator = ['1', '2', '3']
|
||
>>> list(prepend(value, iterator))
|
||
['0', '1', '2', '3']
|
||
|
||
To prepend multiple values, see :func:`itertools.chain`
|
||
or :func:`value_chain`.
|
||
|
||
"""
|
||
return chain([value], iterator)
|
||
|
||
|
||
def convolve(signal, kernel):
|
||
"""Convolve the iterable *signal* with the iterable *kernel*.
|
||
|
||
>>> signal = (1, 2, 3, 4, 5)
|
||
>>> kernel = [3, 2, 1]
|
||
>>> list(convolve(signal, kernel))
|
||
[3, 8, 14, 20, 26, 14, 5]
|
||
|
||
Note: the input arguments are not interchangeable, as the *kernel*
|
||
is immediately consumed and stored.
|
||
|
||
"""
|
||
# This implementation intentionally doesn't match the one in the itertools
|
||
# documentation.
|
||
kernel = tuple(kernel)[::-1]
|
||
n = len(kernel)
|
||
window = deque([0], maxlen=n) * n
|
||
for x in chain(signal, repeat(0, n - 1)):
|
||
window.append(x)
|
||
yield _sumprod(kernel, window)
|
||
|
||
|
||
def before_and_after(predicate, it):
|
||
"""A variant of :func:`takewhile` that allows complete access to the
|
||
remainder of the iterator.
|
||
|
||
>>> it = iter('ABCdEfGhI')
|
||
>>> all_upper, remainder = before_and_after(str.isupper, it)
|
||
>>> ''.join(all_upper)
|
||
'ABC'
|
||
>>> ''.join(remainder) # takewhile() would lose the 'd'
|
||
'dEfGhI'
|
||
|
||
Note that the first iterator must be fully consumed before the second
|
||
iterator can generate valid results.
|
||
"""
|
||
it = iter(it)
|
||
transition = []
|
||
|
||
def true_iterator():
|
||
for elem in it:
|
||
if predicate(elem):
|
||
yield elem
|
||
else:
|
||
transition.append(elem)
|
||
return
|
||
|
||
# Note: this is different from itertools recipes to allow nesting
|
||
# before_and_after remainders into before_and_after again. See tests
|
||
# for an example.
|
||
remainder_iterator = chain(transition, it)
|
||
|
||
return true_iterator(), remainder_iterator
|
||
|
||
|
||
def triplewise(iterable):
|
||
"""Return overlapping triplets from *iterable*.
|
||
|
||
>>> list(triplewise('ABCDE'))
|
||
[('A', 'B', 'C'), ('B', 'C', 'D'), ('C', 'D', 'E')]
|
||
|
||
"""
|
||
# This deviates from the itertools documentation reciple - see
|
||
# https://github.com/more-itertools/more-itertools/issues/889
|
||
t1, t2, t3 = tee(iterable, 3)
|
||
next(t3, None)
|
||
next(t3, None)
|
||
next(t2, None)
|
||
return zip(t1, t2, t3)
|
||
|
||
|
||
def _sliding_window_islice(iterable, n):
|
||
# Fast path for small, non-zero values of n.
|
||
iterators = tee(iterable, n)
|
||
for i, iterator in enumerate(iterators):
|
||
next(islice(iterator, i, i), None)
|
||
return zip(*iterators)
|
||
|
||
|
||
def _sliding_window_deque(iterable, n):
|
||
# Normal path for other values of n.
|
||
it = iter(iterable)
|
||
window = deque(islice(it, n - 1), maxlen=n)
|
||
for x in it:
|
||
window.append(x)
|
||
yield tuple(window)
|
||
|
||
|
||
def sliding_window(iterable, n):
|
||
"""Return a sliding window of width *n* over *iterable*.
|
||
|
||
>>> list(sliding_window(range(6), 4))
|
||
[(0, 1, 2, 3), (1, 2, 3, 4), (2, 3, 4, 5)]
|
||
|
||
If *iterable* has fewer than *n* items, then nothing is yielded:
|
||
|
||
>>> list(sliding_window(range(3), 4))
|
||
[]
|
||
|
||
For a variant with more features, see :func:`windowed`.
|
||
"""
|
||
if n > 20:
|
||
return _sliding_window_deque(iterable, n)
|
||
elif n > 2:
|
||
return _sliding_window_islice(iterable, n)
|
||
elif n == 2:
|
||
return pairwise(iterable)
|
||
elif n == 1:
|
||
return zip(iterable)
|
||
else:
|
||
raise ValueError(f'n should be at least one, not {n}')
|
||
|
||
|
||
def subslices(iterable):
|
||
"""Return all contiguous non-empty subslices of *iterable*.
|
||
|
||
>>> list(subslices('ABC'))
|
||
[['A'], ['A', 'B'], ['A', 'B', 'C'], ['B'], ['B', 'C'], ['C']]
|
||
|
||
This is similar to :func:`substrings`, but emits items in a different
|
||
order.
|
||
"""
|
||
seq = list(iterable)
|
||
slices = starmap(slice, combinations(range(len(seq) + 1), 2))
|
||
return map(operator.getitem, repeat(seq), slices)
|
||
|
||
|
||
def polynomial_from_roots(roots):
|
||
"""Compute a polynomial's coefficients from its roots.
|
||
|
||
>>> roots = [5, -4, 3] # (x - 5) * (x + 4) * (x - 3)
|
||
>>> polynomial_from_roots(roots) # x^3 - 4 * x^2 - 17 * x + 60
|
||
[1, -4, -17, 60]
|
||
"""
|
||
poly = [1]
|
||
for root in roots:
|
||
poly = list(convolve(poly, (1, -root)))
|
||
return poly
|
||
|
||
|
||
def iter_index(iterable, value, start=0, stop=None):
|
||
"""Yield the index of each place in *iterable* that *value* occurs,
|
||
beginning with index *start* and ending before index *stop*.
|
||
|
||
|
||
>>> list(iter_index('AABCADEAF', 'A'))
|
||
[0, 1, 4, 7]
|
||
>>> list(iter_index('AABCADEAF', 'A', 1)) # start index is inclusive
|
||
[1, 4, 7]
|
||
>>> list(iter_index('AABCADEAF', 'A', 1, 7)) # stop index is not inclusive
|
||
[1, 4]
|
||
|
||
The behavior for non-scalar *values* matches the built-in Python types.
|
||
|
||
>>> list(iter_index('ABCDABCD', 'AB'))
|
||
[0, 4]
|
||
>>> list(iter_index([0, 1, 2, 3, 0, 1, 2, 3], [0, 1]))
|
||
[]
|
||
>>> list(iter_index([[0, 1], [2, 3], [0, 1], [2, 3]], [0, 1]))
|
||
[0, 2]
|
||
|
||
See :func:`locate` for a more general means of finding the indexes
|
||
associated with particular values.
|
||
|
||
"""
|
||
seq_index = getattr(iterable, 'index', None)
|
||
if seq_index is None:
|
||
# Slow path for general iterables
|
||
it = islice(iterable, start, stop)
|
||
for i, element in enumerate(it, start):
|
||
if element is value or element == value:
|
||
yield i
|
||
else:
|
||
# Fast path for sequences
|
||
stop = len(iterable) if stop is None else stop
|
||
i = start - 1
|
||
try:
|
||
while True:
|
||
yield (i := seq_index(value, i + 1, stop))
|
||
except ValueError:
|
||
pass
|
||
|
||
|
||
def sieve(n):
|
||
"""Yield the primes less than n.
|
||
|
||
>>> list(sieve(30))
|
||
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
|
||
"""
|
||
if n > 2:
|
||
yield 2
|
||
start = 3
|
||
data = bytearray((0, 1)) * (n // 2)
|
||
limit = math.isqrt(n) + 1
|
||
for p in iter_index(data, 1, start, limit):
|
||
yield from iter_index(data, 1, start, p * p)
|
||
data[p * p : n : p + p] = bytes(len(range(p * p, n, p + p)))
|
||
start = p * p
|
||
yield from iter_index(data, 1, start)
|
||
|
||
|
||
def _batched(iterable, n, *, strict=False):
|
||
"""Batch data into tuples of length *n*. If the number of items in
|
||
*iterable* is not divisible by *n*:
|
||
* The last batch will be shorter if *strict* is ``False``.
|
||
* :exc:`ValueError` will be raised if *strict* is ``True``.
|
||
|
||
>>> list(batched('ABCDEFG', 3))
|
||
[('A', 'B', 'C'), ('D', 'E', 'F'), ('G',)]
|
||
|
||
On Python 3.13 and above, this is an alias for :func:`itertools.batched`.
|
||
"""
|
||
if n < 1:
|
||
raise ValueError('n must be at least one')
|
||
it = iter(iterable)
|
||
while batch := tuple(islice(it, n)):
|
||
if strict and len(batch) != n:
|
||
raise ValueError('batched(): incomplete batch')
|
||
yield batch
|
||
|
||
|
||
if hexversion >= 0x30D00A2:
|
||
from itertools import batched as itertools_batched
|
||
|
||
def batched(iterable, n, *, strict=False):
|
||
return itertools_batched(iterable, n, strict=strict)
|
||
|
||
else:
|
||
batched = _batched
|
||
|
||
batched.__doc__ = _batched.__doc__
|
||
|
||
|
||
def transpose(it):
|
||
"""Swap the rows and columns of the input matrix.
|
||
|
||
>>> list(transpose([(1, 2, 3), (11, 22, 33)]))
|
||
[(1, 11), (2, 22), (3, 33)]
|
||
|
||
The caller should ensure that the dimensions of the input are compatible.
|
||
If the input is empty, no output will be produced.
|
||
"""
|
||
return _zip_strict(*it)
|
||
|
||
|
||
def reshape(matrix, cols):
|
||
"""Reshape the 2-D input *matrix* to have a column count given by *cols*.
|
||
|
||
>>> matrix = [(0, 1), (2, 3), (4, 5)]
|
||
>>> cols = 3
|
||
>>> list(reshape(matrix, cols))
|
||
[(0, 1, 2), (3, 4, 5)]
|
||
"""
|
||
return batched(chain.from_iterable(matrix), cols)
|
||
|
||
|
||
def matmul(m1, m2):
|
||
"""Multiply two matrices.
|
||
|
||
>>> list(matmul([(7, 5), (3, 5)], [(2, 5), (7, 9)]))
|
||
[(49, 80), (41, 60)]
|
||
|
||
The caller should ensure that the dimensions of the input matrices are
|
||
compatible with each other.
|
||
"""
|
||
n = len(m2[0])
|
||
return batched(starmap(_sumprod, product(m1, transpose(m2))), n)
|
||
|
||
|
||
def _factor_pollard(n):
|
||
# Return a factor of n using Pollard's rho algorithm
|
||
gcd = math.gcd
|
||
for b in range(1, n - 2):
|
||
x = y = 2
|
||
d = 1
|
||
while d == 1:
|
||
x = (x * x + b) % n
|
||
y = (y * y + b) % n
|
||
y = (y * y + b) % n
|
||
d = gcd(x - y, n)
|
||
if d != n:
|
||
return d
|
||
raise ValueError('prime or under 5')
|
||
|
||
|
||
_primes_below_211 = tuple(sieve(211))
|
||
|
||
|
||
def factor(n):
|
||
"""Yield the prime factors of n.
|
||
|
||
>>> list(factor(360))
|
||
[2, 2, 2, 3, 3, 5]
|
||
|
||
Finds small factors with trial division. Larger factors are
|
||
either verified as prime with ``is_prime`` or split into
|
||
smaller factors with Pollard's rho algorithm.
|
||
"""
|
||
|
||
# Corner case reduction
|
||
if n < 2:
|
||
return
|
||
|
||
# Trial division reduction
|
||
for prime in _primes_below_211:
|
||
while not n % prime:
|
||
yield prime
|
||
n //= prime
|
||
|
||
# Pollard's rho reduction
|
||
primes = []
|
||
todo = [n] if n > 1 else []
|
||
for n in todo:
|
||
if n < 211**2 or is_prime(n):
|
||
primes.append(n)
|
||
else:
|
||
fact = _factor_pollard(n)
|
||
todo += (fact, n // fact)
|
||
yield from sorted(primes)
|
||
|
||
|
||
def polynomial_eval(coefficients, x):
|
||
"""Evaluate a polynomial at a specific value.
|
||
|
||
Example: evaluating x^3 - 4 * x^2 - 17 * x + 60 at x = 2.5:
|
||
|
||
>>> coefficients = [1, -4, -17, 60]
|
||
>>> x = 2.5
|
||
>>> polynomial_eval(coefficients, x)
|
||
8.125
|
||
"""
|
||
n = len(coefficients)
|
||
if n == 0:
|
||
return x * 0 # coerce zero to the type of x
|
||
powers = map(pow, repeat(x), reversed(range(n)))
|
||
return _sumprod(coefficients, powers)
|
||
|
||
|
||
def sum_of_squares(it):
|
||
"""Return the sum of the squares of the input values.
|
||
|
||
>>> sum_of_squares([10, 20, 30])
|
||
1400
|
||
"""
|
||
return _sumprod(*tee(it))
|
||
|
||
|
||
def polynomial_derivative(coefficients):
|
||
"""Compute the first derivative of a polynomial.
|
||
|
||
Example: evaluating the derivative of x^3 - 4 * x^2 - 17 * x + 60
|
||
|
||
>>> coefficients = [1, -4, -17, 60]
|
||
>>> derivative_coefficients = polynomial_derivative(coefficients)
|
||
>>> derivative_coefficients
|
||
[3, -8, -17]
|
||
"""
|
||
n = len(coefficients)
|
||
powers = reversed(range(1, n))
|
||
return list(map(operator.mul, coefficients, powers))
|
||
|
||
|
||
def totient(n):
|
||
"""Return the count of natural numbers up to *n* that are coprime with *n*.
|
||
|
||
>>> totient(9)
|
||
6
|
||
>>> totient(12)
|
||
4
|
||
"""
|
||
for prime in set(factor(n)):
|
||
n -= n // prime
|
||
return n
|
||
|
||
|
||
# Miller–Rabin primality test: https://oeis.org/A014233
|
||
_perfect_tests = [
|
||
(2047, (2,)),
|
||
(9080191, (31, 73)),
|
||
(4759123141, (2, 7, 61)),
|
||
(1122004669633, (2, 13, 23, 1662803)),
|
||
(2152302898747, (2, 3, 5, 7, 11)),
|
||
(3474749660383, (2, 3, 5, 7, 11, 13)),
|
||
(18446744073709551616, (2, 325, 9375, 28178, 450775, 9780504, 1795265022)),
|
||
(
|
||
3317044064679887385961981,
|
||
(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41),
|
||
),
|
||
]
|
||
|
||
|
||
@lru_cache
|
||
def _shift_to_odd(n):
|
||
'Return s, d such that 2**s * d == n'
|
||
s = ((n - 1) ^ n).bit_length() - 1
|
||
d = n >> s
|
||
assert (1 << s) * d == n and d & 1 and s >= 0
|
||
return s, d
|
||
|
||
|
||
def _strong_probable_prime(n, base):
|
||
assert (n > 2) and (n & 1) and (2 <= base < n)
|
||
|
||
s, d = _shift_to_odd(n - 1)
|
||
|
||
x = pow(base, d, n)
|
||
if x == 1 or x == n - 1:
|
||
return True
|
||
|
||
for _ in range(s - 1):
|
||
x = x * x % n
|
||
if x == n - 1:
|
||
return True
|
||
|
||
return False
|
||
|
||
|
||
def is_prime(n):
|
||
"""Return ``True`` if *n* is prime and ``False`` otherwise.
|
||
|
||
>>> is_prime(37)
|
||
True
|
||
>>> is_prime(3 * 13)
|
||
False
|
||
>>> is_prime(18_446_744_073_709_551_557)
|
||
True
|
||
|
||
This function uses the Miller-Rabin primality test, which can return false
|
||
positives for very large inputs. For values of *n* below 10**24
|
||
there are no false positives. For larger values, there is less than
|
||
a 1 in 2**128 false positive rate. Multiple tests can further reduce the
|
||
chance of a false positive.
|
||
"""
|
||
if n < 17:
|
||
return n in {2, 3, 5, 7, 11, 13}
|
||
if not (n & 1 and n % 3 and n % 5 and n % 7 and n % 11 and n % 13):
|
||
return False
|
||
for limit, bases in _perfect_tests:
|
||
if n < limit:
|
||
break
|
||
else:
|
||
bases = [randrange(2, n - 1) for i in range(64)]
|
||
return all(_strong_probable_prime(n, base) for base in bases)
|
||
|
||
|
||
def loops(n):
|
||
"""Returns an iterable with *n* elements for efficient looping.
|
||
Like ``range(n)`` but doesn't create integers.
|
||
|
||
>>> i = 0
|
||
>>> for _ in loops(5):
|
||
... i += 1
|
||
>>> i
|
||
5
|
||
|
||
"""
|
||
return repeat(None, n)
|